Convergence of Fourier truncations for compact quantum groups and finitely generated groups

TL;DR

This paper extends the concept of Fourier truncations from classical groups to compact quantum groups, demonstrating convergence of these truncations to the original algebra in quantum metric spaces, with applications to quantum groups like SU_q(2).

Contribution

It generalizes Fourier truncation techniques to compact quantum groups and establishes their convergence in quantum Gromov-Hausdorff distance, broadening the scope of noncommutative harmonic analysis.

Findings

Fourier truncations form filtrations of C*-algebras.

Under suitable conditions, these filtrations converge to the original algebra.

Applications include quantum groups SU_q(2) and quantum spheres S^2_q.

Abstract

We generalize the Fej\'er-Riesz operator systems defined for the circle group by Connes and van Suijlekom to the setting of compact matrix quantum groups and their ergodic actions on C*-algebras. These truncations form filtrations of the containing C*-algebra. We show that when they and the containing C*-algebra are equipped with suitable quantum metrics, then under suitable conditions they converge to the containing C*-algebra for quantum Gromov-Hausdorff distance. Among other examples, our results are applicable to the quantum groups $SU_q(2)$ and their homogeneous spaces $S^2_q$.